WORD PROBLEMS WITH/WITHOUT IRRELEVANT INFORMATION

Running Head: WORD PROBLEMS WITH/WITHOUT IRRELEVANT INFORMATION 1

Cognitive and Linguistic Predictors of Mathematical Word Problems With and Without Irrelevant Information

Amber Y. Wanga, Lynn S. Fuchsb, and Douglas Fuchsc Published October 27, 2016

Learning and Individual Differences

aVanderbilt University, Department of Special Education 230 Appleton Place Peabody Box 228 Nashville, TN 37203, USA amber.y.wang@vanderbilt.edu (corresponding author) bVanderbilt University, Department of Special Education lynn.fuchs@vanderbilt.edu cVanderbilt University, Department of Special Education doug.fuchs@vanderbilt.edu This research was supported by Award Number R24HD075443, R01 HD059179, and Core Grant #HD15052 from the Eunice Kennedy Shriver National Institute of Child Health & Human Development to Vanderbilt University. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Eunice Kennedy Shriver National Institute of Child Health & Human Development or the National Institutes of Health.

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Abstract The purpose of this study was to identify cognitive and linguistic predictors of word problems with versus without irrelevant information. The sample was 701 2nd-grade students who received no specialized intervention on word problems. In the fall, they were assessed on initial arithmetic and word-problem skill as well as language ability, working memory capacity, and processing speed; in the spring, they were tested on a word-problem measure that included items with versus without irrelevant information. Significant predictors common to both forms of word problems were initial arithmetic and word problem-solving skill as well as language and working memory. Nonverbal reasoning predicted word problems with irrelevant information, but not word problems without irrelevant information. Findings are discussed in terms of implications for intervention and future research.

Keywords: cognitive predictors, linguistic predictors, problem solving, word problems, irrelevant information

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Cognitive and Linguistic Predictors of Mathematical Word Problems With and Without Irrelevant Information

1. Introduction Word problems (WPs) represent a major component of the mathematics curriculum across kindergarten through high school, and many high-stakes standardized tests, such as the National Assessment of Educational Progress (NAEP; National Assessment Governing Board, 2009), place heavy emphasis on mathematical word-problem solving. This makes sense because among school-age math measures, WPs are the best predictor of adult employment and wages (Every Child a Chance Trust, 2009; Murnane et al., 2001; Parsons & Bynner, 1997; RiveraBatiz, 1992). Therefore, improving word-problem solving is critical for school and occupational success. Not surprisingly, students at risk for or with mathematics learning disabilities struggle with word-problem solving (Parmar, Cawley, & Frazita, 1996). More surprising is that this struggle often occurs in the presence of adequate arithmetic skill (Fuchs et al., 2008; Swanson, Jerman, & Zheng, 2008). Some research indicates that arithmetic and word-problem solving are distinct components of mathematical competence. For example, studies demonstrate that the cognitive and linguistic processes underlying word-problem solving differ from those involved in arithmetic (e.g., Swanson & Beebe-Frankenberger, 2004; Fuchs et al, 2005; Fuchs et al., 2006; Geary et al., 2012). For these reasons, early screening and intervention procedures for preventing WP difficultly likely require a different approach than is needed for arithmetic. Toward this end, understanding the WP features that create challenge is critical. In the present study, we focused on one potentially critical WP feature: the presence of irrelevant information. Specifically, we

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examined whether the cognitive and linguistic student characteristics that predict WP solution accuracy differ for WPs with versus without irrelevant information. We focused on second grade because individual differences in word-problem solving are established at this time (Fuchs et al., 2013) and because second grade is often when identification of learning disability begins (Fletcher, Lyon, Fuchs, & Barnes, 2006).

In this introduction, we begin by providing background information on word-problem solving, including a brief discussion about why word-problem solving may represent challenge in the presence of adequate arithmetic skill and about which cognitive and linguistic factors distinguish between word-problem solving and arithmetic skill. We then turn our attention to complex WPs, specifically those with irrelevant information, and provide a rationale for the present study's focus. 1.1 Word-problem solving: A distinct area of mathematical competence

A major distinction between word-problem solving and arithmetic is the addition of linguistic information, which requires students to decipher a WP narrative in order to build a problem model, identify the missing information, construct a number sentence to find the missing information, and (only then) perform calculation procedures to find the missing number. By contrast, arithmetic problems are already set up for calculation.

Four large-scale studies have examined whether arithmetic and word-problem solving skills constitute a single ability or are distinct areas of mathematical competence, by examining whether the cognitive and linguistic factors underlying word-problem solving and arithmetic differ. In these studies, simple WPs were defined as linguistically presented one-step problems that required arithmetic solutions. Studying 353 first, second, and third graders, Swanson and Beebe-Frankenberger (2004) found that short-term memory and fluid intelligence were unique to

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simple WPs, whereas phonological processing was unique to arithmetic. Working memory contributed strongly to both areas. Fuchs et al. (2005) measured cognitive abilities at the beginning of first grade to predict the development of arithmetic and simple WP skill among 272 children. Common predictors were teacher ratings of attentive behavior and working memory. Nonverbal problem solving was unique to simple WPs and phonological processing was unique to arithmetic. With 312 third graders, Fuchs, Fuchs, Compton, et al. (2006) examined the cognitive correlates of arithmetic versus simple WPs while controlling for the role of arithmetic skill in simple WPs. For simple WPs, nonverbal problem solving, sight word efficiency, language, and concept formation were unique, whereas for arithmetic, processing speed and phonological decoding were unique. Only teacher ratings of attentive behavior were common to both word-problem solving and arithmetic.

With a representative sample of 924 third graders classified as having difficulty with arithmetic, word-problem solving, both domains, or neither, Fuchs, Fuchs, Stuebing et al. (2008) explored patterns of difficulty in arithmetic and word-problem solving. Students were assessed on three measures of word-problem solving and three measures of arithmetic skill, as well as nine cognitive/linguistic dimensions. Using multivariate profile analyses, Fuchs et al. found that specific arithmetic difficulty was associated with deficits in processing speed and attentive behavior and strengths in language. By contrast, word-problem solving was associated with deficits in language. Across these studies, results suggest that individual differences in wordproblem solving are associated with a distinctive set of cognitive and linguistic abilities. 1.2 Sources of WP difficulty

As mentioned, the most transparent distinction between word-problem solving and arithmetic is inclusion of linguistic information. The presentation of linguistic information, or the

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